Create a new bookmark in the panel and copy/paste the line above in the URL field of the Bookmarks panel.

Give it a name like e.g. “GET the POST search”.

Optional add a shortcut like “getit” too.

Now you can go to one of the search engine pages linked above and either click on the bookmarklet or type the shortcut in the URL bar and hit enter.

An alert-box with additional instructions will pop up. Follow the instructions.

Warning: The site will be altered by the bookmarklet. Don’t use it for a search before you have reloaded the page or else it might break!

How it works

The bookmarklet changes all FORM elements on the page to use GET instead of POST. This allows right click > “Add as Search Engine” to detect it and enter the corresponding values it needs. Don’t forget to set the “Use POST Method” checkbox for the newly created search in “Vivaldi > Settings > Search”.

The code being executed is only a few lines so If they went through the trouble of adding the option to check the box for “POST in the search settings, why didn’t they just complete the ability for it’s execution by implementing that code to run automatically if the box is checked so we don’t have to jump through hoops for this to work?

Well, my code is far from perfect, it is a quick hack.
It changes the page (that’s why you really should reload before you interact with the page after using it), which might not be allowed on some sites because of content security policies the site has set, it might break other things which could lead to the browser being banned from entering the site because of “hacking” (and yes, it would be a kind of hacking), the resulting search might not work (I have seen such sites too), the final post request might be composed from the search field plus some stuff that is generated by Javascript (which fails with the bookmarklet), and so on, and so on. I don’t even need to think hard to find a bunch of other things where it might fail or have detrimental effects.

As author of a bookmarklet I don’t have to think about all of that because the user of the bookmarklet uses it at its own risk, but all of this makes the integration in a final product much more difficult than just “fixing” the problem with the bookmarklet.